// Numbas version: exam_results_page_options
{"name": "Tutorial 2 - Bending moments and shear forces", "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "50", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], []], "questions": [{"name": "Determine the loadings", "extensions": [], "custom_part_types": [], "resources": [["question-resources/tutorial2_59M2sS9.svg", "/srv/numbas/media/question-resources/tutorial2_59M2sS9.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Shear and moment diagrams are given for the beam structure shown in the figure below. Determine the loadings on the beam. (Legth: m, Point loads: kN, Distributed loads: kN/m)</p>\n<p></p>\n<p></p>", "advice": "<p>Point $A$:</p>\n<p style=\"padding-left: 30px;\">The jump in the shear force at $A$ corresponds to the vertical reaction at $A$, so $R_A=\\var{ra}$ kN. The moment at $A$ is zero: no applied concentrated moments at $A$.</p>\n<p>Along $AB$:</p>\n<p style=\"padding-left: 30px;\">The shear force is constant ($dV_{AB}/dx=0$), which means there is no distributed load applied along $AB$ ($dV_{AB}/dx=-q_{AB}=0$). The moment is a linear function, whose gradient is equal to the shear force ($dM_{AB}/dx=V_{AB}=const$).</p>\n<p>Point $B$:&nbsp;</p>\n<p style=\"padding-left: 30px;\">There is a jump of $-\\var{P}$kN in the shear force, which indicates that there is a concentration force of $\\var{P}\\mbox{kN}\\downarrow$ applied at p. $B$. The moment distribution is continuious: no concentrated moments at $B$. \\[P_B=\\var{P}\\mbox{ kN}.\\blacktriangleleft\\]</p>\n<p>Along $BC$:</p>\n<p style=\"padding-left: 30px;\">Same argument as for $AB$.</p>\n<p>Point $C$:</p>\n<p style=\"padding-left: 30px;\">Both shear force and moment diagrams are continous at $C$, no jumps: no concentrated forces and moments at $C$.</p>\n<p>Along $CD$:</p>\n<p style=\"padding-left: 30px;\">The shear force is a linear function (straight line); its gradient is constant and negative,&nbsp;which indicated that there is a uniformly distributed load acting on $CD$, whose magnitude can be determined by calculating the gradient of the function $V(x)$ ($dV/dx=-q$):<br/>\\[q_{CD}=-\\frac{dV_{CD}}{dx}=-\\frac{-\\var{rb}-(\\var{rm})}{\\var{a}}=\\var{q}\\mbox{ kN/m}.\\,\\downarrow\\,\\blacktriangleleft\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "2*qa+6*random(1..5#2)", "description": "", "templateType": "anything", "can_override": false}, "qa": {"name": "qa", "group": "Ungrouped variables", "definition": "6*random(3..10)", "description": "", "templateType": "anything", "can_override": false}, "rb": {"name": "rb", "group": "Ungrouped variables", "definition": "(p+5*qa/2)/3.0", "description": "", "templateType": "anything", "can_override": false}, "ra": {"name": "ra", "group": "Ungrouped variables", "definition": "-rb+p+qa", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random([2,3,4,6])", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "qa/a*1.0", "description": "", "templateType": "anything", "can_override": false}, "rm": {"name": "rm", "group": "Ungrouped variables", "definition": "ra-p", "description": "", "templateType": "anything", "can_override": false}, "mb": {"name": "mb", "group": "Ungrouped variables", "definition": "ra*a", "description": "", "templateType": "anything", "can_override": false}, "mc": {"name": "mc", "group": "Ungrouped variables", "definition": "mb+rm*a", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["qa", "p", "rb", "ra", "a", "q", "rm", "mb", "mc"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><img src=\"resources/question-resources/tutorial2_59M2sS9.svg\" width=\"400\" height=\"265\"/></p>\n<p></p>\n<table width=\"625\" height=\"399\" border=\"1\" style=\"height: 244px; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td style=\"width: 138.636px;\"></td>\n<td style=\"width: 104.091px;\">Point load (kN)</td>\n<td style=\"width: 152.273px;\">Point moment (kN.m)</td>\n<td style=\"width: 165px;\">Distributed load (kN/m)</td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>Along AB:</p>\n</td>\n<td style=\"width: 104.091px;\"></td>\n<td style=\"width: 152.273px;\"></td>\n<td style=\"width: 165px;\">[[0]]</td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>At point B:</p>\n</td>\n<td style=\"width: 104.091px;\">[[1]]</td>\n<td style=\"width: 152.273px;\">[[2]]</td>\n<td style=\"width: 165px;\"></td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>Along BC:</p>\n</td>\n<td style=\"width: 104.091px;\"></td>\n<td style=\"width: 152.273px;\"></td>\n<td style=\"width: 165px;\">[[3]]</td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>At point C:</p>\n</td>\n<td style=\"width: 104.091px;\">[[4]]</td>\n<td style=\"width: 152.273px;\">[[5]]</td>\n<td style=\"width: 165px;\"></td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>Along CD:</p>\n</td>\n<td style=\"width: 104.091px;\"></td>\n<td style=\"width: 152.273px;\"></td>\n<td style=\"width: 165px;\">[[6]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "qab", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, 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["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "fc", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mc", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "qcd", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q}", "maxValue": "{q}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Determine the loadings 2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/tutorial2-2_IqVL8h0.svg", "/srv/numbas/media/question-resources/tutorial2-2_IqVL8h0.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Shear and moment diagrams are given for the beam structure shown in the figure below. Determine the loadings on the beam. (Legth: m, Point loads: N, Point moments: N.m, Distributed loads: N/m)</p>", "advice": "<p>See the previous question.</p>\n<p>Point $A$:</p>\n<p style=\"padding-left: 30px;\">The jump in the shear force diagram implies there is a concentrated force in the downward direction \\[P_A=\\var{-ra}\\mbox{ N}.\\,\\downarrow\\,\\blacktriangleleft\\]</p>\n<p style=\"padding-left: 30px;\">The positive jump in the moment diagram means there&nbsp; is a concentrated moment in the clockwise direction \\[M_A=\\var{ma}\\mbox{ Nm}.\\curvearrowright\\,\\blacktriangleleft\\]</p>\n<p>Along $AB$:</p>\n<p style=\"padding-left: 30px;\">No applied loads.</p>\n<p>Point $B$:</p>\n<p style=\"padding-left: 30px;\">No applied loads.</p>\n<p>Along $BC$:</p>\n<p style=\"padding-left: 30px;\">The shear force is a linear function, its gradient is constant \\[q_{BC}=-\\frac{dV_{BC}}{dx}=-\\frac{0-(\\var{ra})}{\\var{a2}}=-\\var{q}\\,\\mbox{ N/m}\\uparrow\\,\\blacktriangleleft\\]</p>\n<p>Point $C$:&nbsp;</p>\n<p style=\"padding-left: 30px;\">The shear force is continuous at $C$, no concentrated forces. The bending moment distribution has a jump of $\\var{-mc}\\mbox{ Nm}$ downwards, which indicates that a concentrated moment is applied at $C$ in the counterclockwise direction.\\[M_C=\\var{mc}\\mbox{ Nm}.\\curvearrowleft\\,\\blacktriangleleft\\]</p>\n<p>Along $CD$:</p>\n<p style=\"padding-left: 30px;\">Same as $BC$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "-ra", "description": "", "templateType": "anything", "can_override": false}, "qa": {"name": "qa", "group": "Ungrouped variables", "definition": "(rb-ra)", "description": "", "templateType": "anything", "can_override": false}, "rb": {"name": "rb", "group": "Ungrouped variables", "definition": "-ra*a3/a2*1.0", "description": "", "templateType": "anything", "can_override": false}, "ra": {"name": "ra", "group": "Ungrouped variables", "definition": "random(-200..-500#-100)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "5", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "qa/a1*1.0", "description": "", "templateType": "anything", "can_override": false}, "mb": {"name": "mb", "group": "Ungrouped variables", "definition": "ma+ra*a1", "description": "", "templateType": "anything", "can_override": false}, "mc": {"name": "mc", "group": "Ungrouped variables", "definition": "random(-100..-500#-100)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "a1-a2", "description": "", "templateType": "anything", "can_override": false}, "mcl": {"name": "mcl", "group": "Ungrouped variables", "definition": "mb+ra*a2/2", "description": "", "templateType": "anything", "can_override": false}, "ma": {"name": "ma", "group": "Ungrouped variables", "definition": "random(200..600#200)", "description": "", "templateType": "anything", "can_override": false}, "mcr": {"name": "mcr", "group": "Ungrouped variables", "definition": "mcl+mc", "description": "", "templateType": "anything", "can_override": false}, "md": {"name": "md", "group": "Ungrouped variables", "definition": "mcr+rb*a3/2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ma", "qa", "ra", "mc", "p", "rb", "a1", "a2", "a3", "q", "mb", "mcl", "mcr", "md"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><img src=\"resources/question-resources/tutorial2-2_IqVL8h0.svg\" width=\"400\" height=\"348\"/>\u200b</p>\n<table width=\"625\" height=\"399\" border=\"1\" style=\"height: 244px; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td style=\"width: 138.636px;\"></td>\n<td style=\"width: 104.091px;\">Point load (N)</td>\n<td style=\"width: 152.273px;\">Point moment (N.m)</td>\n<td style=\"width: 165px;\">Distributed load (N/m)</td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>At point A:</p>\n</td>\n<td style=\"width: 104.091px;\">[[0]]</td>\n<td style=\"width: 152.273px;\">[[1]]</td>\n<td style=\"width: 165px;\"></td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>Along AB:</p>\n</td>\n<td style=\"width: 104.091px;\"></td>\n<td style=\"width: 152.273px;\"></td>\n<td style=\"width: 165px;\">[[2]]</td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>At point B:</p>\n</td>\n<td style=\"width: 104.091px;\">[[3]]</td>\n<td style=\"width: 152.273px;\">[[4]]</td>\n<td style=\"width: 165px;\"></td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>Along BC:</p>\n</td>\n<td style=\"width: 104.091px;\"></td>\n<td style=\"width: 152.273px;\"></td>\n<td style=\"width: 165px;\">[[5]]</td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>At point C:</p>\n</td>\n<td style=\"width: 104.091px;\">[[6]]</td>\n<td style=\"width: 152.273px;\">[[7]]</td>\n<td style=\"width: 165px;\"></td>\n</tr>\n<tr>\n<td style=\"width: 138.636px;\">\n<p>Along CD:</p>\n</td>\n<td style=\"width: 104.091px;\"></td>\n<td style=\"width: 152.273px;\"></td>\n<td style=\"width: 165px;\">[[8]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "fa", "marks": 1, "scripts": {}, 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"{ma}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "qab", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "fb", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", 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"showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mc", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{mc}", "maxValue": "{mc}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "qcd", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-{q}", "maxValue": "-{q}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "M(x) and V(x)", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/figure2P9-0.png", "/srv/numbas/media/question-resources/figure2P9-0.png"], ["question-resources/udl-1_DZLwkPV.svg", "/srv/numbas/media/question-resources/udl-1_DZLwkPV.svg"], ["question-resources/udl-2_sGlnPNr.svg", "/srv/numbas/media/question-resources/udl-2_sGlnPNr.svg"], ["question-resources/udl-3_3fUie75.svg", "/srv/numbas/media/question-resources/udl-3_3fUie75.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Express the shear force $V(x)$ and moment $M(x)$ as a function of $x$ (measured from point $O$), for example \"2x+3\" or \"3*x^2+2*x-1\" or \"5x^3-12*2*x-3*5\".&nbsp;The functions of $x$ you enter are plotted on the graphs. (Note: positive shear force $V$ and moment $M$&nbsp;should follow&nbsp;the agreed&nbsp;<a href=\"resources/question-resources/figure2P9-0.png\" target=\"_blank\" rel=\"noopener noreferrer\">sign convention</a>). Submit your answers, when you are satisfied with the graphs.</p>\n<p><strong><em>Given</em></strong>: $L=\\var{L}$m, $w=\\var{w}$kN/m</p>", "advice": "<p>Using the differential relation $\\frac{dV}{dx}=-w$:</p>\n<p>Along $OB$ ($0 \\le x \\le \\var{L/2}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dV}{dx}=-\\var{w},\\quad\\Rightarrow\\quad V(x)&amp;=&amp;V(0)+\\int_0^x (-\\var{w})dx\\\\&amp;=&amp;0-\\var{w}x=-\\var{w}x.\\blacktriangleleft<br/>\\end{eqnarray*}\\]</p>\n<p>and along $BC$ ($\\var{L/2}\\le x \\le \\var{L}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dV}{dx}=0,\\quad\\Rightarrow\\quad V(x)&amp;=&amp;V(\\var{L/2})+\\int_{x=\\var{L/2}}^x 0\\, dx\\\\<br/>&amp;=&amp; V(\\var{L/2})=-\\var{w}\\times \\var{L/2}=-\\var{w*L/2}.\\blacktriangleleft\\end{eqnarray*}\\]</p>\n<p>Now to get the moment as a function of $x$:</p>\n<p>Along $OB$ ($0 \\le x \\le \\var{L/2}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dM}{dx}=V(x)=-\\var{w}x,\\quad\\Rightarrow\\quad M(x)&amp;=&amp;M(0)+\\int_0^x (-\\var{w}x)\\,dx\\\\<br/>&amp;=&amp; 0\\mbox{(free end)}-\\var{w}\\frac{x^2}{2}=-\\var{w/2} x^2.\\blacktriangleleft\\end{eqnarray*}\\]</p>\n<p>and along $BC$ ($\\var{L/2}\\le x \\le \\var{L}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dM}{dx}=V(x)=-\\var{w*L/2},\\quad\\Rightarrow\\quad M(x)&amp;=&amp;M(\\var{L/2})+\\int_{\\var{L/2}}^x (-\\var{w*L/2})\\,dx\\\\<br/>&amp;=&amp; -\\var{w*L^2/8}-\\var{w*L/2}(x-\\var{L/2})=\\var{w*L^2/8}-\\var{w*L/2}x.\\blacktriangleleft\\end{eqnarray*}\\]</p>", "rulesets": {}, "variables": {"w": {"name": "w", "group": "Ungrouped variables", "definition": "random(30..60#2)", "description": "", "templateType": "anything"}, "shear_gaps": {"name": "shear_gaps", "group": "Ungrouped variables", "definition": "[0,1]", "description": "", "templateType": "anything"}, "moment_gaps": {"name": "moment_gaps", "group": "Ungrouped variables", "definition": "[2,3]", "description": "", "templateType": "anything"}, "L": {"name": "L", "group": "Ungrouped variables", "definition": "random(2..10#2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["w", "shear_gaps", "moment_gaps", "L"], "variable_groups": [], "functions": {"plot": {"parameters": [["w", "number"], ["my_gaps", "list"], ["label", "string"], ["L", "number"]], "type": "html", "language": "javascript", "definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n  \n// The line is described by the equation \n//   y = a*x + b \n\n// This function takes as its parameters the coefficients a and b,\n// and the coordinates (x2,y2) of a point on the line.\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar len=L;\nvar half=L/2;\n\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox: [-3,w*len^2,1.5*len,-w*len^2],\n showNavigation: true,\n axis:true\n});\n  \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board;  \n\n// create the x-axis.\n// var xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\n// var xticks = board.create('ticks',[xaxis,2],{\n//   drawLabels: true,\n//   label: {offset: [-4, -10]},\n//   minorTicks: 0\n// });\n\n// create the y-axis\n// var yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\n// var yticks = board.create('ticks',[yaxis,100],{\n// drawLabels: true,\n// label: {offset: [-30, 0]},\n// minorTicks: 0\n//});\n\n//var shade_range = function(f,x1,x2) {\n//  var dataX1 = [x1,x1];\n//  var dataY1 = [0,f(x1)];\n//  var dataX2 = [];\n // var dataY2 = [];\n // for (var i = x1; i <= x2; i = i+1) {\n//      dataX2.push(i);\n//      dataY2.push(f(i));\n//  }\n\n//  var dataX3 = [x2,x2];\n//  var dataY3 = [f(x2),0];\n\n  //var dataX = dataX1.concat(dataX2).concat(dataX3);\n  //var dataY = dataY1.concat(dataY2).concat(dataY3);\n\n  //var dataxy=[dataX,dataY];\n//return dataxy\n//}\n\n                                    \nboard.create('text',[1, w*8,label]);\n// create the static line based on the coefficients a and b\n//var line1 = board.create('line',[[0,b],[1,a+b]],{fixed:true, strokeWidth: 1});\n\n// mark the two given points - one on the y-axis, and one at (x2,y2)\n//var p1 = board.create('point',[0,b],{fixed:true, size:3, name: 'P_1', face: 'cross'});\n//var p2 = board.create('point',[x2,y2],{fixed:true, size:3, name: 'P_2', face: 'cross'});\n\n// Now we can do the clever stuff with the student's answer!\n// We'll add a curve to the board which is a plot of a function we provide.\n// That function will parse the student's input and evaluate it.\n\n// The variable `studentExpression` will store the parsed version of\n// the student's expression.\nvar studentExpression_AB;\n\n// This function evaluates the student's expression at a given point `t`.\nvar makestudentline_AB=function(x){\n  // Create a JME scope with the variable x set to the given value.\n  var nscope = new Numbas.jme.Scope([Numbas.jme.builtinScope,{variables: {x: new Numbas.jme.types.TNum(x)}}]);\n  \n  // If the student's input has been parsed, evaluate it\n  if(studentExpression_AB) {\n    try {\n  var val = Numbas.jme.evaluate(studentExpression_AB,nscope).value;\n  return val;}\ncatch(e) {  \n  // If there was an error evaluating the student's expression\n  // (wrong variables, or some other weirdness) throw an error\n  throw(e)\n}\n  }\n  // Otherwise, if the student's expression hasn't been parsed\n  // (they haven't written anything, or they wrote bad syntax) return 0\n  else {\nreturn 0;\n  }\n}\n\nvar studentExpression_BC;\n\n// This function evaluates the student's expression at a given point `x`.\nvar makestudentline_BC=function(x){\n  // Create a JME scope with the variable x set to the given value.\n  var nscope = new Numbas.jme.Scope([Numbas.jme.builtinScope,\n                                     {variables: {x: new Numbas.jme.types.TNum(x)}}]);\n  \n  // If the student's input has been parsed, evaluate it\n  if(studentExpression_BC) {\ntry {\n  var val = Numbas.jme.evaluate(studentExpression_BC,nscope).value;\n  return val;\n}\ncatch(e) {\n  // If there was an error evaluating the student's expression\n  // (wrong variables, or some other weirdness)\n  // throw an error\n  throw(e)\n}\n  }\n  // Otherwise, if the student's expression hasn't been parsed\n  // (they haven't written anything, or they wrote bad syntax)\n  // return 0\n  else {\nreturn 0;\n  }\n};\n\nvar studentline_AB = board.create('functiongraph', \n   [makestudentline_AB,0,half],\n   {strokeColor:'blue', strokeWidth: 3, visible: false}\n  );\nvar studentline_BC = board.create('functiongraph', \n   [makestudentline_BC,half,len],\n   {strokeColor:'blue', strokeWidth: 3, visible: false}\n  );\nvar studentline=function(x){if (x<half) {return makestudentline_AB(x);} \n                            else {return makestudentline_BC(x);};};\n\nvar myplot=board.create('functiongraph',[studentline,0,len],{strokeColor:'blue', strokeWidth: 3,visible:false});  \n//var sh =board.create('curve',shade_range(studentline,0,len),\n //                          {strokeWidth:1, fillColor:'red', fillOpacity:0.3,visible:true});//   \n\n\n\n//var sh_AB=board.create('integral',[[0.01,6],studentline_AB],{withlabel:false,fillOpacity:0.2, visible: false});\n//var sh_BC=board.create('integral',[[6,12],studentline_BC],{withlabel:false,fillOpacity:0.2, visible: false});\n//var sh_AB=shade(makestudentline_AB,0,6,'red');\n//var sh_BC=shade(makestudentline_BC,6,12,'red');\n//var sh_AB =board.create('curve',shade_range(makestudentline_AB,0,6),\n//                            {strokeWidth:0, fillColor:'red', fillOpacity:0.2,visible:false});//\n//var sh_BC =board.create('curve',shade_range(makestudentline_BC,6,12),\n//                            {strokeWidth:0, fillColor:'red', fillOpacity:0.2,visible:false});//\n\n//var studentarea_AB = board.create('integral',[[6,-0.0],studentline_AB],{visible: false});\n//var studentarea_BC = board.create('integral', [[6,12],studentline_BC], {visible: false});\n// This is where some voodoo happens.\n// Because the HTML for the question is inserted into the page after the function eqnline\n// is called, we need to wait until the 'question-html-attached' event is fired\n// to do the interaction with the student input box.\n// So:\n\n// When the question is inserted into the page\n//question.signals.on('HTMLAttached',function(e) {\n  \n// Create a Knockout.js observable\n\nko.computed(function(){\n// Get the student's input string from part 0, gap 0.\nvar studentString_AB = question.parts[0].gaps[my_gaps[0]].display.studentAnswer();\nvar studentString_BC = question.parts[0].gaps[my_gaps[1]].display.studentAnswer();\n// Try to parse it as a JME expression\ntry {\n  studentExpression_AB = Numbas.jme.compile(studentString_AB,scope);\n  studentExpression_BC = Numbas.jme.compile(studentString_BC,scope);\n\n  // If the student didn't write anything, compile returns null\n//if(studentExpression_AB === null)\n//throw(new Error('no expression'));\n//   if(studentExpression_BC === null)\n//throw(new Error('no expression'));\n// If everything worked, show the line and update it\n// (this calls makestudentline_AB on a few points)\n//  studentline_AB.showElement();\n//  studentline_AB.updateCurve();\n//  studentline_BC.showElement();\n//  studentline_BC.updateCurve();\n// board.suspendUpdate();\n  myplot.showElement();\nmyplot.updateCurve();\n//sh.showElement();\n//sh.updateRenderer();\n\n  \n//var leftline1=board.create('segment',[[0,0],[0,function(){return makestudentline_AB(0)}]]);\n//var rightline1=board.create('segment',[[6,0],[6,function(){return makestudentline_AB(6)}]]);\n//var leftline2=board.create('segment',[[6,0],[6,function(){return makestudentline_BC(6)}]]);\n//var rightline2=board.create('segment',[[12,0],[12,function(){return makestudentline_BC(12)}]]);\n //board.unsuspendUpdate();\n \n\n}\ncatch(e) {\n  // If something went wrong, hide the curve\n  studentExpression_AB = null;\n  //studentline_AB.hideElement();\n  studentExpression_BC = null;\n  //studentline_BC.hideElement();\n}\n\nboard.update();\n//var  g = board.create('glider', [myplot]);//g.showElement();    g.showElement();\n//var graphint = board.create('integral',[[0,12],myplot],\n    //{withlabel:false, fixed:true,curveLeft: {visible: false},curveRight: {visible: false},visible: false});\n})\n\n//});  \n\nreturn div;\n\n\n"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\"><img src=\"resources/question-resources/udl-2_sGlnPNr.svg\" width=\"404\" height=\"106\"/></p>\n<p style=\"text-align: left;\"><strong>Shear force:</strong></p>\n<p style=\"text-align: left; padding-left: 30px;\">Along $OB$ ($0\\le x \\le \\var{L/2}$):&nbsp; &nbsp; $V(x)=$[[0]], kN<br/>Along $BC$ ($\\var{L/2}\\le x \\le \\var{L}$):&nbsp;&nbsp;&nbsp;$V(x)=$[[1]], kN</p>\n<p style=\"text-align: left; padding-left: 30px;\">{plot(w,shear_gaps,'V, kN',L)}</p>\n<p><strong>Bending moment:</strong></p>\n<p style=\"padding-left: 30px;\">Along $OB$ ($0\\le x \\le \\var{L/2}$):&nbsp; $M(x)=$[[2]], kNm<br/>Along $BC$ ($\\var{L/2}\\le x \\le \\var{L}$):&nbsp;$M(x)=$[[3]], kNm</p>\n<p style=\"padding-left: 30px;\">{plot(w,moment_gaps,'M, kNm',L)}</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{w} x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{w*L/2.0}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{w/2.0}x^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{w*L^2/8.0}-{w*L/2.0}x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "M(x) and V(x) - v2", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/figure2P9-0.png", "/srv/numbas/media/question-resources/figure2P9-0.png"], ["question-resources/udl-1_DZLwkPV.svg", "/srv/numbas/media/question-resources/udl-1_DZLwkPV.svg"], ["question-resources/udl-2_sGlnPNr.svg", "/srv/numbas/media/question-resources/udl-2_sGlnPNr.svg"], ["question-resources/udl-3_3fUie75.svg", "/srv/numbas/media/question-resources/udl-3_3fUie75.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Express the shear force $V(x)$ and moment $M(x)$ as a function of $x$ (measured from point $O$), for example \"2x+3\" or \"3*x^2+2*x-1\" or \"5x^3-12*2*x-3*5\".&nbsp;The functions of $x$ you enter are plotted on the graphs. (Note: positive shear force $V$ and moment $M$&nbsp;should follow the agreed&nbsp;<a href=\"resources/question-resources/figure2P9-0.png\" target=\"_blank\" rel=\"noopener noreferrer\">sign convention</a>). Submit your answers, when you are satisfied with the graphs.</p>\n<p><strong><em>Given</em></strong>: $L=\\var{L}$m, $w=\\var{w}$kN/m, $P=\\var{P}$kN.</p>", "advice": "<p>Using the differential relation $\\frac{dV}{dx}=-w$:</p>\n<p>Along $OB$ ($0&nbsp;&lt; x&nbsp;&lt; \\var{L/2}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dV}{dx}=-\\var{w},\\quad\\Rightarrow\\quad V_{OB}(x)&amp;=&amp;V(0)+\\int_0^x (-\\var{w})dx\\\\&amp;=&amp;0-\\var{w}x=-\\var{w}x.\\blacktriangleleft<br/>\\end{eqnarray*}\\]</p>\n<p>and along $BC$ ($\\var{L/2} &lt; x&nbsp;&lt; \\var{L}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dV_{BC}}{dx}=0,\\quad\\Rightarrow\\quad V_{BC}(x)&amp;=&amp;V(\\var{L/2})+\\int_{\\var{L/2}}^x 0\\, dx\\\\<br/>&amp;=&amp; V_{OB}(\\var{L/2})-P=-\\var{w}\\times \\var{L/2}-\\var{P}=-\\var{w*L/2+P}.\\blacktriangleleft\\end{eqnarray*}\\]</p>\n<p>Notice that you need to add the concentrated force $P$, which acts at p. $B$. In the shear force diagram there will be a jump in the direction of $P$.</p>\n<p>Now to get the moment as a function of $x$:</p>\n<p>Along $OB$ ($0&nbsp;&lt; x&nbsp;&lt; \\var{L/2}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dM_{OB}}{dx}=V_{OB}(x)=-\\var{w}x,\\quad\\Rightarrow\\quad M_{OB}(x)&amp;=&amp;M(0)+\\int_0^x (-\\var{w}x)\\,dx\\\\<br/>&amp;=&amp; 0\\mbox{(free end)}-\\var{w}\\frac{x^2}{2}=-\\var{w/2} x^2.\\blacktriangleleft\\end{eqnarray*}\\]</p>\n<p>and along $BC$ ($\\var{L/2} &lt; x&nbsp;&lt; \\var{L}$):</p>\n<p>\\[\\begin{eqnarray*}\\frac{dM_{BC}}{dx}=V_{BC}(x)=-\\var{w*L/2+P},\\quad\\Rightarrow\\quad M_{BC}(x)&amp;=&amp;M(\\var{L/2})+\\int_{\\var{L/2}}^x (-\\var{w*L/2+P})\\,dx\\\\<br/>&amp;=&amp; -\\var{w*L^2/8}\\mbox{ (no external moments at B)}-\\var{w*L/2+P}(x-\\var{L/2})=\\var{w*L^2/8+P*L/2}-\\var{w*L/2+P}x.\\blacktriangleleft\\end{eqnarray*}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"w": {"name": "w", "group": "Ungrouped variables", "definition": "random(30..60#2)", "description": "", "templateType": "anything", "can_override": false}, "shear_gaps": {"name": "shear_gaps", "group": "Ungrouped variables", "definition": "[0,1]", "description": "", "templateType": "anything", "can_override": false}, "moment_gaps": {"name": "moment_gaps", "group": "Ungrouped variables", "definition": "[2,3]", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Ungrouped variables", "definition": "random(2..10#2)", "description": "", "templateType": "anything", "can_override": false}, "P": {"name": "P", "group": "Ungrouped variables", "definition": "random(20..60#10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["w", "shear_gaps", "moment_gaps", "L", "P"], "variable_groups": [], "functions": {"plot": {"parameters": [["w", "number"], ["my_gaps", "list"], ["label", "string"], ["L", "number"]], "type": "html", "language": "javascript", "definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n  \n// The line is described by the equation \n//   y = a*x + b \n\n// This function takes as its parameters the coefficients a and b,\n// and the coordinates (x2,y2) of a point on the line.\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar len=L;\nvar half=L/2;\n\nvar div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',\n{boundingBox: [-3,w*len^2,1.5*len,-w*len^2],\n showNavigation: true,\n axis:true\n});\n  \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board;  \n\n// create the x-axis.\n// var xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\n// var xticks = board.create('ticks',[xaxis,2],{\n//   drawLabels: true,\n//   label: {offset: [-4, -10]},\n//   minorTicks: 0\n// });\n\n// create the y-axis\n// var yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\n// var yticks = board.create('ticks',[yaxis,100],{\n// drawLabels: true,\n// label: {offset: [-30, 0]},\n// minorTicks: 0\n//});\n\n//var shade_range = function(f,x1,x2) {\n//  var dataX1 = [x1,x1];\n//  var dataY1 = [0,f(x1)];\n//  var dataX2 = [];\n // var dataY2 = [];\n // for (var i = x1; i <= x2; i = i+1) {\n//      dataX2.push(i);\n//      dataY2.push(f(i));\n//  }\n\n//  var dataX3 = [x2,x2];\n//  var dataY3 = [f(x2),0];\n\n  //var dataX = dataX1.concat(dataX2).concat(dataX3);\n  //var dataY = dataY1.concat(dataY2).concat(dataY3);\n\n  //var dataxy=[dataX,dataY];\n//return dataxy\n//}\n\n                                    \nboard.create('text',[1, w*8,label]);\n// create the static line based on the coefficients a and b\n//var line1 = board.create('line',[[0,b],[1,a+b]],{fixed:true, strokeWidth: 1});\n\n// mark the two given points - one on the y-axis, and one at (x2,y2)\n//var p1 = board.create('point',[0,b],{fixed:true, size:3, name: 'P_1', face: 'cross'});\n//var p2 = board.create('point',[x2,y2],{fixed:true, size:3, name: 'P_2', face: 'cross'});\n\n// Now we can do the clever stuff with the student's answer!\n// We'll add a curve to the board which is a plot of a function we provide.\n// That function will parse the student's input and evaluate it.\n\n// The variable `studentExpression` will store the parsed version of\n// the student's expression.\nvar studentExpression_AB;\n\n// This function evaluates the student's expression at a given point `t`.\nvar makestudentline_AB=function(x){\n  // Create a JME scope with the variable x set to the given value.\n  var nscope = new Numbas.jme.Scope([Numbas.jme.builtinScope,{variables: {x: new Numbas.jme.types.TNum(x)}}]);\n  \n  // If the student's input has been parsed, evaluate it\n  if(studentExpression_AB) {\n    try {\n  var val = Numbas.jme.evaluate(studentExpression_AB,nscope).value;\n  return val;}\ncatch(e) {  \n  // If there was an error evaluating the student's expression\n  // (wrong variables, or some other weirdness) throw an error\n  throw(e)\n}\n  }\n  // Otherwise, if the student's expression hasn't been parsed\n  // (they haven't written anything, or they wrote bad syntax) return 0\n  else {\nreturn 0;\n  }\n}\n\nvar studentExpression_BC;\n\n// This function evaluates the student's expression at a given point `x`.\nvar makestudentline_BC=function(x){\n  // Create a JME scope with the variable x set to the given value.\n  var nscope = new Numbas.jme.Scope([Numbas.jme.builtinScope,\n                                     {variables: {x: new Numbas.jme.types.TNum(x)}}]);\n  \n  // If the student's input has been parsed, evaluate it\n  if(studentExpression_BC) {\ntry {\n  var val = Numbas.jme.evaluate(studentExpression_BC,nscope).value;\n  return val;\n}\ncatch(e) {\n  // If there was an error evaluating the student's expression\n  // (wrong variables, or some other weirdness)\n  // throw an error\n  throw(e)\n}\n  }\n  // Otherwise, if the student's expression hasn't been parsed\n  // (they haven't written anything, or they wrote bad syntax)\n  // return 0\n  else {\nreturn 0;\n  }\n};\n\nvar studentline_AB = board.create('functiongraph', \n   [makestudentline_AB,0,half],\n   {strokeColor:'blue', strokeWidth: 3, visible: false}\n  );\nvar studentline_BC = board.create('functiongraph', \n   [makestudentline_BC,half,len],\n   {strokeColor:'blue', strokeWidth: 3, visible: false}\n  );\nvar studentline=function(x){if (x<half) {return makestudentline_AB(x);} \n                            else {return makestudentline_BC(x);};};\n\nvar myplot=board.create('functiongraph',[studentline,0,len],{strokeColor:'blue', strokeWidth: 3,visible:false});  \n//var sh =board.create('curve',shade_range(studentline,0,len),\n //                          {strokeWidth:1, fillColor:'red', fillOpacity:0.3,visible:true});//   \n\n\n\n//var sh_AB=board.create('integral',[[0.01,6],studentline_AB],{withlabel:false,fillOpacity:0.2, visible: false});\n//var sh_BC=board.create('integral',[[6,12],studentline_BC],{withlabel:false,fillOpacity:0.2, visible: false});\n//var sh_AB=shade(makestudentline_AB,0,6,'red');\n//var sh_BC=shade(makestudentline_BC,6,12,'red');\n//var sh_AB =board.create('curve',shade_range(makestudentline_AB,0,6),\n//                            {strokeWidth:0, fillColor:'red', fillOpacity:0.2,visible:false});//\n//var sh_BC =board.create('curve',shade_range(makestudentline_BC,6,12),\n//                            {strokeWidth:0, fillColor:'red', fillOpacity:0.2,visible:false});//\n\n//var studentarea_AB = board.create('integral',[[6,-0.0],studentline_AB],{visible: false});\n//var studentarea_BC = board.create('integral', [[6,12],studentline_BC], {visible: false});\n// This is where some voodoo happens.\n// Because the HTML for the question is inserted into the page after the function eqnline\n// is called, we need to wait until the 'question-html-attached' event is fired\n// to do the interaction with the student input box.\n// So:\n\n// When the question is inserted into the page\n//question.signals.on('HTMLAttached',function(e) {\n  \n// Create a Knockout.js observable\n\nko.computed(function(){\n// Get the student's input string from part 0, gap 0.\nvar studentString_AB = question.parts[0].gaps[my_gaps[0]].display.studentAnswer();\nvar studentString_BC = question.parts[0].gaps[my_gaps[1]].display.studentAnswer();\n// Try to parse it as a JME expression\ntry {\n  studentExpression_AB = Numbas.jme.compile(studentString_AB,scope);\n  studentExpression_BC = Numbas.jme.compile(studentString_BC,scope);\n\n  // If the student didn't write anything, compile returns null\n//if(studentExpression_AB === null)\n//throw(new Error('no expression'));\n//   if(studentExpression_BC === null)\n//throw(new Error('no expression'));\n// If everything worked, show the line and update it\n// (this calls makestudentline_AB on a few points)\n//  studentline_AB.showElement();\n//  studentline_AB.updateCurve();\n//  studentline_BC.showElement();\n//  studentline_BC.updateCurve();\n// board.suspendUpdate();\n  myplot.showElement();\nmyplot.updateCurve();\n//sh.showElement();\n//sh.updateRenderer();\n\n  \n//var leftline1=board.create('segment',[[0,0],[0,function(){return makestudentline_AB(0)}]]);\n//var rightline1=board.create('segment',[[6,0],[6,function(){return makestudentline_AB(6)}]]);\n//var leftline2=board.create('segment',[[6,0],[6,function(){return makestudentline_BC(6)}]]);\n//var rightline2=board.create('segment',[[12,0],[12,function(){return makestudentline_BC(12)}]]);\n //board.unsuspendUpdate();\n \n\n}\ncatch(e) {\n  // If something went wrong, hide the curve\n  studentExpression_AB = null;\n  //studentline_AB.hideElement();\n  studentExpression_BC = null;\n  //studentline_BC.hideElement();\n}\n\nboard.update();\n//var  g = board.create('glider', [myplot]);//g.showElement();    g.showElement();\n//var graphint = board.create('integral',[[0,12],myplot],\n    //{withlabel:false, fixed:true,curveLeft: {visible: false},curveRight: {visible: false},visible: false});\n})\n\n//});  \n\nreturn div;\n\n\n"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\"><img src=\"resources/question-resources/udl-3_3fUie75.svg\" width=\"403\" height=\"136\"/></p>\n<p style=\"text-align: left;\"><strong>Shear force:</strong></p>\n<p style=\"text-align: left; padding-left: 30px;\">Along $OB$ ($0\\le x \\le \\var{L/2}$):&nbsp; &nbsp; $V(x)=$[[0]], kN<br/>Along $BC$ ($\\var{L/2}\\le x \\le \\var{L}$):&nbsp;&nbsp;&nbsp;$V(x)=$[[1]], kN</p>\n<p style=\"text-align: left; padding-left: 30px;\">{plot(w,shear_gaps,'V, kN',L)}</p>\n<p><strong>Bending moment:</strong></p>\n<p style=\"padding-left: 30px;\">Along $OB$ ($0\\le x \\le \\var{L/2}$):&nbsp; $M(x)=$[[2]], kNm<br/>Along $BC$ ($\\var{L/2}\\le x \\le \\var{L}$):&nbsp;$M(x)=$[[3]], kNm</p>\n<p style=\"padding-left: 30px;\">{plot(w,moment_gaps,'M, kNm',L)}</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{w} x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{w*L/2.0+P}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{w/2.0}x^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{w*L^2/8.0+P*L/2}-{w*L/2.0+P}x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Max moment and shear force", "extensions": [], "custom_part_types": [], "resources": [["question-resources/udl-4.svg", "/srv/numbas/media/question-resources/udl-4.svg"], ["question-resources/udl-5.svg", "/srv/numbas/media/question-resources/udl-5.svg"], ["question-resources/udl-15_Ozi07hp.svg", "/srv/numbas/media/question-resources/udl-15_Ozi07hp.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>For the beam shown in the figure below, determine the magintude of the maximum shear force and bending moment, if:</p>\n<p><img src=\"resources/question-resources/udl-4.svg\" width=\"459\" height=\"147\"/></p>", "advice": "<p><strong>(a)&nbsp;</strong>$L=\\var{L}$ m, $P=\\var{P1}$ kN and $w=\\var{w}$ kN/m.</p>\n<ol>\n<li>Determine the reactions at the supports:<br/>\\[R_{\\text{left}}=\\frac{2}{3}P+\\frac{1}{18}wL=\\var{2*P1/3+w*L/18} \\mbox{ kN},\\uparrow\\]<br/>\\[R_{\\text{right}}=\\frac{1}{3}P+\\frac{5}{18}wL=\\var{P1/3+5*w*L/18} \\mbox{ kN}.\\uparrow\\]</li>\n<li>Draw the shear force and bending moment diagrams:<br/><br/><img src=\"resources/question-resources/udl-5.svg\"/><br/><br/></li>\n<li>The maximum shear force is&nbsp;\\[|V_{\\max}|=\\var{v_max_1}\\mbox{ kN}.\\,\\blacktriangleleft\\]</li>\n</ol>\n<p style=\"padding-left: 60px;\">an dthe maximum bending moment is \\[M_{\\max}=\\var{m_max_1}\\mbox{ kNm}.\\,\\blacktriangleleft\\]</p>\n<p><strong>(b)&nbsp;</strong>$L=\\var{L}$ m, $P=\\var{P2}$ kN and $w=\\var{w}$ kN/m.</p>\n<ol>\n<li>Determine the reactions at the supports:<br/>\\[R_A=\\frac{2}{3}P+\\frac{1}{18}wL=\\var{2*P2/3+w*L/18} \\mbox{ kN},\\uparrow\\]<br/>\\[R_B=\\frac{1}{3}P+\\frac{5}{18}wL=\\var{P2/3+5*w*L/18} \\mbox{ kN}.\\uparrow\\]</li>\n<li>The shear force and bending moment diagrams:<br/><br/><img src=\"resources/question-resources/udl-15_Ozi07hp.svg\"/><br/><br/><br/></li>\n<li>The maximum shear force is&nbsp;\\[|V_{\\max}|=\\var{v_max_2},\\mbox{ kN}.\\,\\blacktriangleleft\\]</li>\n<li>Firstly, determine the location&nbsp; $x^*$ of the maximum bending moment: <br/>\\[\\frac{dM(x^*)}{dx}=V(x^*)=0,\\quad\\Rightarrow\\,\\, \\var{vb2}-\\var{w}(x^*-\\var{2*L/3})=0,\\]<br/>Solve for $x^*$:<br/>\\[x^*=\\frac{\\var{vb2}+\\var{w}\\times\\var{2*L/3}}{\\var{w}}=\\var{x_m_max_2}\\mbox{m}.\\]<br/>Now, calculate the moment at&nbsp;$x^*$. It&nbsp;can be done by looking at equilibrium of the beam part to the right of $x^*$:<br/>\\[M(x^*)=R_{\\text{right}}(L-x^*)+\\frac{w(L-x^*)^2}{2}=\\var{m_max_2}\\mbox{ kNm}.\\blacktriangleleft\\]<br/>Alternatively, we can integrate the shear force from $x=\\var{2*L/3}$ to $x^*$:<br/>\\[M(x^*)=\\var{mc2}+\\frac{1}{2}\\var{vb2}(\\var{x_m_max_2}-\\var{2*L/3})=\\var{m_max_2}\\mbox{ kNm}.\\blacktriangleleft\\]</li>\n</ol>", "rulesets": {}, "variables": {"L": {"name": "L", "group": "Ungrouped variables", "definition": "random(3..12#3)", "description": "", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "random(9..54#9)*2", "description": "", "templateType": "anything"}, "P1": {"name": "P1", "group": "Ungrouped variables", "definition": "w*L/6.0+random(6..36#6)", "description": "", "templateType": "anything"}, "P2": {"name": "P2", "group": "Ungrouped variables", "definition": "random(6..w*L/8#3)", "description": "", "templateType": "anything"}, "ra1": {"name": "ra1", "group": "Ungrouped variables", "definition": "2*P1/3.0+w*L/18.0", "description": "", "templateType": "anything"}, "ra2": {"name": "ra2", "group": "Ungrouped variables", "definition": "2*P2/3.0+w*L/18.0", "description": "", "templateType": "anything"}, "rb1": {"name": "rb1", "group": "Ungrouped variables", "definition": "P1/3+5*w*L/18", "description": "", "templateType": "anything"}, "rb2": {"name": "rb2", "group": "Ungrouped variables", "definition": "P2/3+5*w*L/18.0", "description": "", "templateType": "anything"}, "mb1": {"name": "mb1", "group": "Ungrouped variables", "definition": "L*(12*P1+w*L)/54", "description": "", "templateType": "anything"}, "mb2": {"name": "mb2", "group": "Ungrouped variables", "definition": "L*(12*P2+w*L)/54.0", "description": "", "templateType": "anything"}, "mc1": {"name": "mc1", "group": "Ungrouped variables", "definition": "L*(3*P1+w*L)/27", "description": "", "templateType": "anything"}, "mc2": {"name": "mc2", "group": "Ungrouped variables", "definition": "L*(3*P2+w*L)/27.0", "description": "", "templateType": "anything"}, "vb1": {"name": "vb1", "group": "Ungrouped variables", "definition": "ra1-P1", "description": "", "templateType": "anything"}, "vb2": {"name": "vb2", "group": "Ungrouped variables", "definition": "ra2-P2", "description": "", "templateType": "anything"}, "v_max_1": {"name": "v_max_1", "group": "Ungrouped variables", "definition": "max(ra1,rb1)", "description": "", "templateType": "anything"}, "m_max_1": {"name": "m_max_1", "group": "Ungrouped variables", "definition": "mb1", "description": "", "templateType": "anything"}, "v_max_2": {"name": "v_max_2", "group": "Ungrouped variables", "definition": "max(ra2,rb2)", "description": "", "templateType": "anything"}, "x_m_max_2": {"name": "x_m_max_2", "group": "Ungrouped variables", "definition": "precround((vb2+w*2*L/3)/w,3)", "description": "<p>Location of the max bending moment/zero shear force</p>", "templateType": "anything"}, "m_max_2": {"name": "m_max_2", "group": "Ungrouped variables", "definition": "precround((6*P2 + 5*L*w)^2/(648*w),2)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "L/3.0", "description": "", "templateType": "anything"}, "m_max": {"name": "m_max", "group": "Ungrouped variables", "definition": "precround(rb2*(L-x_m_max_2)-w*(L-x_m_max_2)^2/2,2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "mc1-mb1<0\nand\nmc2-mb2>0", "maxRuns": "1000"}, "ungrouped_variables": ["L", "a", "w", "P1", "P2", "ra1", "ra2", "rb1", "rb2", "mb1", "mb2", "mc1", "mc2", "vb1", "vb2", "v_max_1", "m_max_1", "v_max_2", "x_m_max_2", "m_max_2", "m_max"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$L=\\var{L}$ m, $P=\\var{P1}$ kN and $w=\\var{w}$ kN/m.</p>\n<p></p>\n<p style=\"padding-left: 30px;\">$|V_{\\max}|=$[[0]] kN,&nbsp;</p>\n<p style=\"padding-left: 30px;\">$|M_{\\max}|=$[[1]] kNm.</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{v_max_1}-0.1", "maxValue": "{v_max_1}+0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m_max_1}-0.1", "maxValue": "{m_max_1}+0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$L=\\var{L}$ m, $P=\\var{P2}$ kN and $w=\\var{w}$ kN/m.</p>\n<p style=\"padding-left: 30px;\">$|V_{\\max}|=$[[0]] kN,</p>\n<p style=\"padding-left: 30px;\">$|M_{\\max}|=$[[1]] kNm.</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{v_max_2}-0.1", "maxValue": "{v_max_2}+0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{m_max_2}-0.1", "maxValue": "{m_max_2}+0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Beam bending: Generic algorithm for V(x) and M(x) in a simply supported beam", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/bending.svg", "/srv/numbas/media/question-resources/bending.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A simply supported steel beam of length&nbsp;$L=\\var{L}$ m&nbsp;is loaded as shown in the figure below, when&nbsp;$P=\\var{P[1]}$ N and&nbsp;$w=\\var{w[0]}$ N/m.</p>\n<p><img src=\"resources/question-resources/bending.svg\" width=\"429\" height=\"133\"/></p>", "advice": "<p>Determine the reactions:</p>\n<p>$R_A=\\var{rA}$ N.</p>\n<p>$R_D=\\var{rB}$ N</p>\n<p><strong>(a)</strong></p>\n<p>Shear&nbsp;force distribution $V(x)$, N (you can hover over&nbsp;a point to see its coordinates)</p>\n<p>{correctv(x,v_left,v_right,vlimits)}</p>\n<p><strong>(b)</strong></p>\n<p>Bending moment distribution $M(x)$, Nm:</p>\n<p>{correctm(x ,m_left,m_right,m_middle,mlimits)}</p>\n<p><strong>(c)</strong></p>\n<p>Since $dM(x)/dx=V(x)$, the maximum bending moment is where $V(x_{cr})=0$.</p>\n<p>Along AB the shear force is $V(x)=R_A-w x$. Hence, the shear force vanishes at&nbsp;</p>\n<p>\\[x_{cr}=\\frac{R_A}{w}=\\var{xcr} \\text{m}. \\blacktriangleleft\\]</p>\n<p>The moment is the integral of the shear force (area under the shear force distribution):</p>\n<p>\\[M(x_{cr})=\\underbrace{M_A}_{=0}+\\frac{1}{2}R_A x_{cr}=0.5\\times \\var{rA}\\times \\var{xcr}=\\var{mmax}\\text{Nm}.\\blacktriangleleft\\]</p>", "rulesets": {}, "variables": {"intervals": {"name": "intervals", "group": "shear force", "definition": "sort(x+x[1..len(x)-1])\n", "description": "", "templateType": "anything"}, "intervalsM": {"name": "intervalsM", "group": "bending moment", "definition": "sort(x[0..len(x)-1]+x[1..len(x)]+map(x[j]+ll[j]/2,j,0..len(ll)-1))", "description": "", "templateType": "anything"}, "P": {"name": "P", "group": "Loads", "definition": "[0,random(150..600#150)*2]//repeat(random(1.2..7#0.4),len(x)-2)", "description": "", "templateType": "anything"}, "w": {"name": "w", "group": "Loads", "definition": "[random(1..4)*300,0,0]//repeat(random(1.2..7#0.4),len(x)-1)", "description": "", "templateType": "anything"}, "L": {"name": "L", "group": "Loads", "definition": "6.0", "description": "", "templateType": "anything"}, "rB": {"name": "rB", "group": "Loads", "definition": "sum(map(P[j]*(x[j+1]),j,list(0..len(p)-1)))/L\n  +sum(map(w[j]*ll[j]*(x[j]+0.5*ll[j]),j,list(0..len(w)-1)))/L\n  +sum(m0)/L", "description": "", "templateType": "anything"}, "rA": {"name": "rA", "group": "Loads", "definition": "sum(map(P[j]*(L-x[j+1]),j,list(0..len(p)-1)))/L\n  +sum(map(w[j]*ll[j]*(L-x[j]-0.5*ll[j]),j,list(0..len(w)-1)))/L\n  -sum(m0)/L", "description": "", "templateType": "anything"}, "vlimits": {"name": "vlimits", "group": "shear force", "definition": "[max(v_left+v_right),min(v_left+v_right)]", "description": "", "templateType": "anything"}, "mlimits": {"name": "mlimits", "group": "bending moment", "definition": "[max(m_left+m_right)+1,if(min(m_left+m_right)=0,-max(m_left+m_right)/4,min(m_left+m_right))]", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Loads", "definition": "[0,L/3,2*L/3,L]", "description": "", "templateType": "anything"}, "equil": {"name": "equil", "group": "Loads", "definition": "[sum(P)+sum(map(w[j]*ll[j],j,list(0..len(w)-1))),rA+rB]", "description": "", "templateType": "anything"}, "all_p": {"name": "all_p", "group": "shear force", "definition": "[ra]+map(-p[j],j,0..len(p)-1)+[rb]", "description": "", "templateType": "anything"}, "ll": {"name": "ll", "group": "Loads", "definition": "map(x[j+1]-x[j],j,0..len(x)-2)", "description": "", "templateType": "anything"}, "all_w": {"name": "all_w", "group": "shear force", "definition": "map(-w[j]*(ll[j]),j,0..len(w)-1)", "description": "", "templateType": "anything"}, "v_left": {"name": "v_left", "group": "shear force", "definition": "map(cumsum(all_p)[j]+cumsum(all_w)[j],j,0..len(w))", "description": "", "templateType": "anything"}, "v_right": {"name": "v_right", "group": "shear force", "definition": "map(cumsum(all_p)[j+1]+cumsum(all_w)[j],j,0..len(w))", "description": "", "templateType": "anything"}, "m_left": {"name": "m_left", "group": "bending moment", "definition": "map(cumsum(all_v)[j]+cumsum(m0)[j],j,0..len(all_v))", "description": "", "templateType": "anything"}, "m_right": {"name": "m_right", "group": "bending moment", "definition": "map(m_left[j]+m0[j],j,0..len(m0)-1)", "description": "", "templateType": "anything"}, "m0": {"name": "m0", "group": "Loads", "definition": "[0,0,0,0]", "description": "", "templateType": "anything"}, "all_v": {"name": "all_v", "group": "bending moment", "definition": "map((v_right[j]+v_left[j+1])*ll[j]/2,j,0..len(v_left)-2)", "description": "", "templateType": "anything"}, "m_middle": {"name": "m_middle", "group": "bending moment", "definition": "map(m_right[j]+(3*v_right[j]+v_left[j+1])*ll[j]/8,j,0..len(ll)-1)", "description": "", "templateType": "anything"}, "xcr": {"name": "xcr", "group": "Max bending moment", "definition": "precround(v_right[0]/w[0],3)", "description": "", "templateType": "anything"}, "mmax": {"name": "mmax", "group": "Max bending moment", "definition": "m0[0]+0.5*v_right[0]*xcr", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "v_right[1]<0", "maxRuns": "500"}, "ungrouped_variables": [], "variable_groups": [{"name": "Loads", "variables": ["L", "x", "ll", "P", "w", "rA", "rB", "equil", "m0"]}, {"name": "shear force", "variables": ["intervals", "all_p", "all_w", "v_left", "v_right", "vlimits"]}, {"name": "bending moment", "variables": ["intervalsM", "all_v", "m_left", "m_right", "mlimits", "m_middle"]}, {"name": "Max bending moment", "variables": ["xcr", "mmax"]}], "functions": {"dragpoint_board": {"parameters": [["intervals", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "\n  var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',{boundingBox:[-1,limits[0]*1.2,intervals[intervals.length-1]+1,\n                                                                               limits[1]*1.2],showNavigation:true});//grid:true,\n //$(question.display.html).find('#dragpoint_board').append(div); //adds  <div> to html\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  //shorthand to evaluate a mathematical expression to a number\n  function evaluate(expression) {\n    try {\n      var val = Numbas.jme.evaluate(expression,question.scope);\n      return Numbas.jme.unwrapValue(val);\n    }\n    catch(e) {\n      // if there's an error, return no number\n      return NaN;\n    }\n  }\n  \n  // set up points array\n  var num_points = intervals.length;\n  var points = [];\n  \n  \n  // this function sets up the i^th point\n  function make_point(i) {\n    \n    // calculate initial coordinates\n    var x = intervals[i];\n    \n    // create an invisible vertical line for the point to slide along\n    var line = board.create('line',[[x,0],[x,1]],{fixed:true,visible:false});\n    \n    // create the point\n    var point = points[i] = board.create(\n      'glider',\n      [intervals[i],0,line],\n      {\n        withlabel:false,//name:'',\n        size:2,\n        snapSizeY: 25,  // the point will snap to y-coordinates which are multiples of 0.1\n        snapToGrid: true\n      }\n    );\n    \n    // the contents of the input box for this point\n    var studentAnswer = question.parts[0].gaps[i].display.studentAnswer;\n    \n    // watch the student's input and reposition the point when it changes.    \n    ko.computed(function() {\n      y = evaluate(studentAnswer());\n      if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n        point.moveTo([x,y],100);\n      }\n    });\n    \n    // when the student drags the point, update the gapfill input\n    point.on('drag',function(){\n      var y = Numbas.math.niceNumber(point.Y());\n      studentAnswer(y);\n    });\n    \n  }\n  \n // create each point\n  for(var i=0;i<num_points;i++) {\n    make_point(i);\n  } \n\n//create dashed lines\nvar unique_points=intervals.unique();\nfor (var i=1;i<num_points-2;i++){\n  //create lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n\n  var p0=board.create('point',[0,0],{fixed:true,withlabel:false});\n  var pL=board.create('point',[intervals[num_points-1],0],{fixed:true,withlabel:false});\n  for(var i=1;i<num_points;i++) {\n    //make_point(i);\n    board.create('segment',[points[i-1],points[i]]);\n//    trial_line=board.create('functiongraph',function(x) {return points[i=1].Y()+(x-points[i-1].X())*(points[i].Y()-points[i-1].Y())/(points[i].X()-points[i-1].X());});  \n// board.create('integral',[[0,1],trial_line]);     \n  }\n\n  \n\n  // create the parabola\n // var parabool = board.create('spline',points,{strokeWidth:3,strokeColor:'red'});\n  board.create('segment',[p0,points[0]]);  \n  board.create('segment',[points[num_points-1],[intervals[num_points-1],0]]);  \n  //var c=board.create('point',[(points[1].X()+points[0].X())/2,(points[1].Y()+points[0].Y())/2]);\n  //var parabola=board.create('spline',[points[0],c,points[1]]);\n\n  return div;"}, "piecewise_moment": {"parameters": [["intervals", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "\n  var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',\n                                                 {boundingBox:[-1,limits[0]*1.2,intervals[intervals.length-1]+1,limits[1]*1.2]});\n //$(question.display.html).find('#piecewise_moment').append(div);\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  //shorthand to evaluate a mathematical expression to a number\n  function evaluate(expression) {\n    try {\n      var val = Numbas.jme.evaluate(expression,question.scope);\n      return Numbas.jme.unwrapValue(val);\n    }\n    catch(e) {\n      // if there's an error, return no number\n      return NaN;\n    }\n  }\n  \n  // set up points array\n  var num_points = intervals.length;\n  var points = [];\n  \n  \n  // this function sets up the i^th point\n  function make_point(i) {\n    \n    // calculate initial coordinates\n    var x = intervals[i];\n    \n    // create an invisible vertical line for the point to slide along\n    var line = board.create('line',[[x,0],[x,1]],{fixed:true,visible:false});\n    \n    // create the point\n    var point = points[i] = board.create(\n      'glider',\n      [intervals[i],0,line],\n      {\n        name:'',\n        size:2,\n        snapSizeY: 25,  // the point will snap to y-coordinates which are multiples of 0.1\n        snapToGrid: true\n      }\n    );\n    \n    // the contents of the input box for this point\n    var studentAnswer = question.parts[1].gaps[i].display.studentAnswer;\n    \n    // watch the student's input and reposition the point when it changes.    \n    ko.computed(function() {\n      y = evaluate(studentAnswer());\n      if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n        point.moveTo([x,y],100);\n      }\n    });\n    \n    // when the student drags the point, update the gapfill input\n    point.on('drag',function(){\n      var y = Numbas.math.niceNumber(point.Y());\n      studentAnswer(y);\n    });\n    \n  }\n  \n // create each point\n  for(var i=0;i<num_points;i++) {\n    make_point(i);\n  } \n\n//create dashed lines\nvar unique_points=intervals.unique();\nfor (var i=0;i<num_points-2;i++){\n  //create lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n//end points\n  var p0=board.create('point',[0,0],{fixed:true,withlabel:false});\n  var pL=board.create('point',[intervals[num_points-1],0],{fixed:true,withlabel:false});\n\n//parabolas  \nfor(var i=0;i<num_points-2;i+=3) {\n  var polyInt = JXG.Math.Numerics.lagrangePolynomial([points[i],points[i+1],points[i+2]]);\n  var gr=board.create('functiongraph',[polyInt,intervals[i],intervals[i+2]]);  \n}\nfor(var i=2;i<num_points-3;i+=3) {board.create('segment',[points[i], points[i+1]]);} ; \n  \n  board.create('segment',[p0,points[0]]);  \n  board.create('segment',[points[num_points-1],[intervals[num_points-1],0]]);  \n\n  return div;"}, "cumsum": {"parameters": [["v", "list of number"]], "type": "list", "language": "jme", "definition": "map(sum(v[0..j]),j,0..len(v))"}, "correctv": {"parameters": [["x", "list of number"], ["vL", "list of number"], ["vR", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',{boundingBox:[-1,limits[0]*1.2,x[x.length-1]+1,limits[1]*1.2],grid:false});\n //$(question.display.html).find('#piecewise_moment').append(div);\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  // set up points array\n  var num_points = x.length;\n  var points = [];\n  \n//create dashed lines\nvar unique_points=x.unique();\nfor (var i=1;i<unique_points.length;i++){\n  //create vertical lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n\nboard.create('point',[0,0],{fixed:true,withlabel:false});\nboard.create('point',[x[x.length-1],0],{fixed:true,withlabel:false});\nboard.create('segment',[[0,0],[0,vR[0]]],{fixed:true});\n\nfor (var i=0;i<unique_points.length-1;i++){\n  p1 = board.create('point',[x[i],vR[i]],{fixed:true,withlabel:false});\n  p2 = board.create('point',[x[i+1],vL[i+1]],{fixed:true,withlabel:false});\n  \n  board.create('segment',[p1,p2]);\n\n  board.create('segment',[p2,[x[i+1],vR[i+1]]]);\n}\n return div;"}, "correctm": {"parameters": [["x", "list of number"], ["mL", "list of number"], ["mR", "list of number"], ["mM", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "//var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',{boundingBox:[-1,limits[0]*1.2,x[x.length-1]+1,limits[1]*1.2],grid:false});\n //$(question.display.html).find('#piecewise_moment').append(div);\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  // set up points array\n  var num_points = x.length;\n  var points = [];\n  \n//create dashed lines\nvar unique_points=x.unique();\nfor (var i=1;i<unique_points.length;i++){\n  //create vertical lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n\nboard.create('point',[0,0],{fixed:true,withlabel:false});\nboard.create('point',[x[x.length-1],0],{fixed:true,withlabel:false});\nboard.create('segment',[[0,0],[0,mR[0]]],{fixed:true});\nvar p=[];\nfor (var i=0;i<x.length-1;i++){\n  p1 = board.create('point',[x[i],mR[i]],{fixed:true,withlabel:false});\n  p2 = board.create('point',[0.5*(x[i]+x[i+1]),mM[i]],{fixed:true,withlabel:false});\n  p3 = board.create('point',[x[i+1],mL[i+1]],{fixed:true,withlabel:false});\n  \n  var polyInt = JXG.Math.Numerics.lagrangePolynomial([p1,p2,p3]);\n  var gr=board.create('functiongraph',[polyInt,x[i],x[i+1]]);  \n  //board.create('segment',[p1,p2]);\n  board.create('segment',[p3,[x[i+1],mR[i+1]]]);\n}\n return div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Fill in the table of values $V(x)$&nbsp;at&nbsp;the end points of each interval. You can also hold and drag the red points on the graph.</p>\n<table border=\"1\">\n<thead>\n<tr>\n<th>Point</th>\n<th>A</th>\n<th>B left</th>\n<th>B right</th>\n<th>C left</th>\n<th>C right</th>\n<th>D</th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td>$V$, N</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n</tr>\n</tbody>\n</table>\n<div></div>\n<div>{dragpoint_board(intervals,vlimits)}</div>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "vA", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_right[0]", "maxValue": "v_right[0]", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vB_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_left[1]", "maxValue": "v_left[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vB_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_right[1]", "maxValue": "v_right[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vC_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_left[2]", "maxValue": "v_left[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vC_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_right[2]", "maxValue": "v_right[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vD", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_left[3]", "maxValue": "v_left[3]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Fill in the table of values for $M(x)$&nbsp;at the end and middle points of each interval. You can also hold and drag the red points on the graph.&nbsp;&nbsp;\u200b</p>\n<table border=\"1\">\n<thead>\n<tr>\n<th>Point</th>\n<th>A</th>\n<th>midle AB</th>\n<th>B left</th>\n<th>B right</th>\n<th>midle BC</th>\n<th>C left</th>\n<th>C right</th>\n<th>midle CD</th>\n<th>D</th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td>$M$, Nm</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n<td>[[8]]</td>\n</tr>\n</tbody>\n</table>\n<div></div>\n<div>{piecewise_moment(intervalsM,mlimits)}</div>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "mA", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[0]", "maxValue": "m_right[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mAB", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_middle[0]", "maxValue": "m_middle[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mB_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_left[1]", "maxValue": "m_left[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mB_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[1]", "maxValue": "m_right[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mBC", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_middle[1]", "maxValue": "m_middle[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mC_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_left[2]", "maxValue": "m_left[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mC_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[2]", "maxValue": "m_right[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mCD", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_middle[2]", "maxValue": "m_middle[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mD", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[3]", "maxValue": "m_right[3]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Determine magnitude of the maximum bending moment (in Nm) and its location $x_{cr}$ measured in m from point A.</p>\n<p style=\"padding-left: 30px;\">$M_\\max=$[[0]]&nbsp;</p>\n<p style=\"padding-left: 30px;\">$x_{cr}=$[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Mmax", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "mmax-3", "maxValue": "mmax+3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "xcr", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "xcr-0.05", "maxValue": "xcr+0.05", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Beam bending 2", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/bending-3.svg", "/srv/numbas/media/question-resources/bending-3.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A simply supported steel beam of length&nbsp;$L=\\var{L}$ m&nbsp;is loaded as shown in the figure below, when&nbsp;$P_1=\\var{P[0]}$ N,&nbsp;$P_2=\\var{P[1]}$ N and&nbsp;$w=\\var{w[1]}$ N/m.</p>\n<p><img src=\"resources/question-resources/bending-3.svg\" width=\"448\" height=\"142\"/></p>", "advice": "<p>Determine the reactions:</p>\n<p>$R_A=\\var{rA}$ N.</p>\n<p>$R_D=\\var{rB}$ N</p>\n<p><strong>(a)</strong></p>\n<p>Shear&nbsp;force distribution $V(x)$, N (you can hover over&nbsp;a point to see its coordinates)</p>\n<p>{correctv(x,v_left,v_right,vlimits)}</p>\n<p><strong>(b)</strong></p>\n<p>Bending moment distribution $M(x)$, Nm:</p>\n<p>{correctm(x ,m_left,m_right,m_middle,mlimits)}</p>\n<p><strong>(c)</strong></p>\n<p>Since $dM(x)/dx=V(x)$, the maximum bending moment is where $V(x_{cr})=0$.</p>\n<p>Along BC the shear force is $V(x)=V_{B}-w x$, when $x$ is measured form B. Hence, the shear force vanishes at&nbsp;</p>\n<p>\\[x_{cr}=\\frac{V_B}{w}=\\var{xcr} \\text{m},\\]</p>\n<p>Distance&nbsp;from A: \\[d=\\var{xcr}+\\var{L/3}=\\var{xcr+L/3}\\text{m}.\\blacktriangleleft\\]</p>\n<p>The moment is the integral of the shear force (area under the shear force distribution):</p>\n<p>\\[M(x_{cr})=M_B+\\frac{1}{2}V_B x_{cr}=\\var{m_right[1]}+0.5\\times \\var{v_right[1]}\\times \\var{xcr}=\\var{mmax}\\text{Nm}.\\blacktriangleleft\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"intervals": {"name": "intervals", "group": "shear force", "definition": "sort(x+x[1..len(x)-1])\n", "description": "", "templateType": "anything", "can_override": false}, "intervalsM": {"name": "intervalsM", "group": "bending moment", "definition": "sort(x[0..len(x)-1]+x[1..len(x)]+map(x[j]+ll[j]/2,j,0..len(ll)-1))", "description": "", "templateType": "anything", "can_override": false}, "P": {"name": "P", "group": "Loads", "definition": "[p1,random(300..1200#300 except p1)]", "description": "", "templateType": "anything", "can_override": false}, "w": {"name": "w", "group": "Loads", "definition": "[0,random(1..4)*300,0]//repeat(random(1.2..7#0.4),len(x)-1)", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Loads", "definition": "6.0", "description": "", "templateType": "anything", "can_override": false}, "rB": {"name": "rB", "group": "Loads", "definition": "sum(map(P[j]*(x[j+1]),j,list(0..len(p)-1)))/L\n  +sum(map(w[j]*ll[j]*(x[j]+0.5*ll[j]),j,list(0..len(w)-1)))/L\n  +sum(m0)/L", "description": "", "templateType": "anything", "can_override": false}, "rA": {"name": "rA", "group": "Loads", "definition": "sum(map(P[j]*(L-x[j+1]),j,list(0..len(p)-1)))/L\n  +sum(map(w[j]*ll[j]*(L-x[j]-0.5*ll[j]),j,list(0..len(w)-1)))/L\n  -sum(m0)/L", "description": "", "templateType": "anything", "can_override": false}, "vlimits": {"name": "vlimits", "group": "shear force", "definition": "[max(v_left+v_right),min(v_left+v_right)]", "description": "", "templateType": "anything", "can_override": false}, "mlimits": {"name": "mlimits", "group": "bending moment", "definition": "[max(m_left+m_right)+1,if(min(m_left+m_right)=0,-max(m_left+m_right)/4,min(m_left+m_right))]", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Loads", "definition": "[0,L/3,2*L/3,L]", "description": "", "templateType": "anything", "can_override": false}, "equil": {"name": "equil", "group": "Loads", "definition": "[sum(P)+sum(map(w[j]*ll[j],j,list(0..len(w)-1))),rA+rB]", "description": "", "templateType": "anything", "can_override": false}, "all_p": {"name": "all_p", "group": "shear force", "definition": "[ra]+map(-p[j],j,0..len(p)-1)+[rb]", "description": "", "templateType": "anything", "can_override": false}, "ll": {"name": "ll", "group": "Loads", "definition": "map(x[j+1]-x[j],j,0..len(x)-2)", "description": "", "templateType": "anything", "can_override": false}, "all_w": {"name": "all_w", "group": "shear force", "definition": "map(-w[j]*(ll[j]),j,0..len(w)-1)", "description": "", "templateType": "anything", "can_override": false}, "v_left": {"name": "v_left", "group": "shear force", "definition": "map(cumsum(all_p)[j]+cumsum(all_w)[j],j,0..len(w))", "description": "", "templateType": "anything", "can_override": false}, "v_right": {"name": "v_right", "group": "shear force", "definition": "map(cumsum(all_p)[j+1]+cumsum(all_w)[j],j,0..len(w))", "description": "", "templateType": "anything", "can_override": false}, "m_left": {"name": "m_left", "group": "bending moment", "definition": "map(cumsum(all_v)[j]+cumsum(m0)[j],j,0..len(all_v))", "description": "", "templateType": "anything", "can_override": false}, "m_right": {"name": "m_right", "group": "bending moment", "definition": "map(m_left[j]+m0[j],j,0..len(m0)-1)", "description": "", "templateType": "anything", "can_override": false}, "m0": {"name": "m0", "group": "Loads", "definition": "[0,0,0,0]", "description": "", "templateType": "anything", "can_override": false}, "all_v": {"name": "all_v", "group": "bending moment", "definition": "map((v_right[j]+v_left[j+1])*ll[j]/2,j,0..len(v_left)-2)", "description": "", "templateType": "anything", "can_override": false}, "m_middle": {"name": "m_middle", "group": "bending moment", "definition": "map(m_right[j]+(3*v_right[j]+v_left[j+1])*ll[j]/8,j,0..len(ll)-1)", "description": "", "templateType": "anything", "can_override": false}, "xcr": {"name": "xcr", "group": "Max bending moment", "definition": "precround(v_right[1]/w[1],3)", "description": "", "templateType": "anything", "can_override": false}, "mmax": {"name": "mmax", "group": "Max bending moment", "definition": "m_right[1]+0.5*v_right[1]*xcr", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Loads", "definition": "random(150..600#150)*2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "v_right[1]*v_left[2]<0", "maxRuns": "500"}, "ungrouped_variables": [], "variable_groups": [{"name": "Loads", "variables": ["L", "x", "ll", "P", "w", "rA", "rB", "equil", "m0", "p1"]}, {"name": "shear force", "variables": ["intervals", "all_p", "all_w", "v_left", "v_right", "vlimits"]}, {"name": "bending moment", "variables": ["intervalsM", "all_v", "m_left", "m_right", "mlimits", "m_middle"]}, {"name": "Max bending moment", "variables": ["xcr", "mmax"]}], "functions": {"dragpoint_board": {"parameters": [["intervals", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "\n  var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',{boundingBox:[-1,limits[0]*1.2,intervals[intervals.length-1]+1,\n                                                                               limits[1]*1.2],showNavigation:true});//grid:true,\n //$(question.display.html).find('#dragpoint_board').append(div); //adds  <div> to html\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  //shorthand to evaluate a mathematical expression to a number\n  function evaluate(expression) {\n    try {\n      var val = Numbas.jme.evaluate(expression,question.scope);\n      return Numbas.jme.unwrapValue(val);\n    }\n    catch(e) {\n      // if there's an error, return no number\n      return NaN;\n    }\n  }\n  \n  // set up points array\n  var num_points = intervals.length;\n  var points = [];\n  \n  \n  // this function sets up the i^th point\n  function make_point(i) {\n    \n    // calculate initial coordinates\n    var x = intervals[i];\n    \n    // create an invisible vertical line for the point to slide along\n    var line = board.create('line',[[x,0],[x,1]],{fixed:true,visible:false});\n    \n    // create the point\n    var point = points[i] = board.create(\n      'glider',\n      [intervals[i],0,line],\n      {\n        withlabel:false,//name:'',\n        size:2,\n        snapSizeY: 25,  // the point will snap to y-coordinates which are multiples of 0.1\n        snapToGrid: true\n      }\n    );\n    \n    // the contents of the input box for this point\n    var studentAnswer = question.parts[0].gaps[i].display.studentAnswer;\n    \n    // watch the student's input and reposition the point when it changes.    \n    ko.computed(function() {\n      y = evaluate(studentAnswer());\n      if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n        point.moveTo([x,y],100);\n      }\n    });\n    \n    // when the student drags the point, update the gapfill input\n    point.on('drag',function(){\n      var y = Numbas.math.niceNumber(point.Y());\n      studentAnswer(y);\n    });\n    \n  }\n  \n // create each point\n  for(var i=0;i<num_points;i++) {\n    make_point(i);\n  } \n\n//create dashed lines\nvar unique_points=intervals.unique();\nfor (var i=1;i<num_points-2;i++){\n  //create lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n\n  var p0=board.create('point',[0,0],{fixed:true,withlabel:false});\n  var pL=board.create('point',[intervals[num_points-1],0],{fixed:true,withlabel:false});\n  for(var i=1;i<num_points;i++) {\n    //make_point(i);\n    board.create('segment',[points[i-1],points[i]]);\n//    trial_line=board.create('functiongraph',function(x) {return points[i=1].Y()+(x-points[i-1].X())*(points[i].Y()-points[i-1].Y())/(points[i].X()-points[i-1].X());});  \n// board.create('integral',[[0,1],trial_line]);     \n  }\n\n  \n\n  // create the parabola\n // var parabool = board.create('spline',points,{strokeWidth:3,strokeColor:'red'});\n  board.create('segment',[p0,points[0]]);  \n  board.create('segment',[points[num_points-1],[intervals[num_points-1],0]]);  \n  //var c=board.create('point',[(points[1].X()+points[0].X())/2,(points[1].Y()+points[0].Y())/2]);\n  //var parabola=board.create('spline',[points[0],c,points[1]]);\n\n  return div;"}, "piecewise_moment": {"parameters": [["intervals", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "\n  var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',\n                                                 {boundingBox:[-1,limits[0]*1.2,intervals[intervals.length-1]+1,limits[1]*1.2]});\n //$(question.display.html).find('#piecewise_moment').append(div);\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  //shorthand to evaluate a mathematical expression to a number\n  function evaluate(expression) {\n    try {\n      var val = Numbas.jme.evaluate(expression,question.scope);\n      return Numbas.jme.unwrapValue(val);\n    }\n    catch(e) {\n      // if there's an error, return no number\n      return NaN;\n    }\n  }\n  \n  // set up points array\n  var num_points = intervals.length;\n  var points = [];\n  \n  \n  // this function sets up the i^th point\n  function make_point(i) {\n    \n    // calculate initial coordinates\n    var x = intervals[i];\n    \n    // create an invisible vertical line for the point to slide along\n    var line = board.create('line',[[x,0],[x,1]],{fixed:true,visible:false});\n    \n    // create the point\n    var point = points[i] = board.create(\n      'glider',\n      [intervals[i],0,line],\n      {\n        name:'',\n        size:2,\n        snapSizeY: 25,  // the point will snap to y-coordinates which are multiples of 0.1\n        snapToGrid: true\n      }\n    );\n    \n    // the contents of the input box for this point\n    var studentAnswer = question.parts[1].gaps[i].display.studentAnswer;\n    \n    // watch the student's input and reposition the point when it changes.    \n    ko.computed(function() {\n      y = evaluate(studentAnswer());\n      if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n        point.moveTo([x,y],100);\n      }\n    });\n    \n    // when the student drags the point, update the gapfill input\n    point.on('drag',function(){\n      var y = Numbas.math.niceNumber(point.Y());\n      studentAnswer(y);\n    });\n    \n  }\n  \n // create each point\n  for(var i=0;i<num_points;i++) {\n    make_point(i);\n  } \n\n//create dashed lines\nvar unique_points=intervals.unique();\nfor (var i=0;i<num_points-2;i++){\n  //create lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n//end points\n  var p0=board.create('point',[0,0],{fixed:true,withlabel:false});\n  var pL=board.create('point',[intervals[num_points-1],0],{fixed:true,withlabel:false});\n\n//parabolas  \nfor(var i=0;i<num_points-2;i+=3) {\n  var polyInt = JXG.Math.Numerics.lagrangePolynomial([points[i],points[i+1],points[i+2]]);\n  var gr=board.create('functiongraph',[polyInt,intervals[i],intervals[i+2]]);  \n}\nfor(var i=2;i<num_points-3;i+=3) {board.create('segment',[points[i], points[i+1]]);} ; \n  \n  board.create('segment',[p0,points[0]]);  \n  board.create('segment',[points[num_points-1],[intervals[num_points-1],0]]);  \n\n  return div;"}, "cumsum": {"parameters": [["v", "list of number"]], "type": "list", "language": "jme", "definition": "map(sum(v[0..j]),j,0..len(v))"}, "correctv": {"parameters": [["x", "list of number"], ["vL", "list of number"], ["vR", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',{boundingBox:[-1,limits[0]*1.2,x[x.length-1]+1,limits[1]*1.2],grid:false});\n //$(question.display.html).find('#piecewise_moment').append(div);\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  // set up points array\n  var num_points = x.length;\n  var points = [];\n  \n//create dashed lines\nvar unique_points=x.unique();\nfor (var i=1;i<unique_points.length;i++){\n  //create vertical lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n\nboard.create('point',[0,0],{fixed:true,withlabel:false});\nboard.create('point',[x[x.length-1],0],{fixed:true,withlabel:false});\nboard.create('segment',[[0,0],[0,vR[0]]],{fixed:true});\n\nfor (var i=0;i<unique_points.length-1;i++){\n  p1 = board.create('point',[x[i],vR[i]],{fixed:true,withlabel:false});\n  p2 = board.create('point',[x[i+1],vL[i+1]],{fixed:true,withlabel:false});\n  \n  board.create('segment',[p1,p2]);\n\n  board.create('segment',[p2,[x[i+1],vR[i+1]]]);\n}\n return div;"}, "correctm": {"parameters": [["x", "list of number"], ["mL", "list of number"], ["mR", "list of number"], ["mM", "list of number"], ["limits", "list of number"]], "type": "html", "language": "javascript", "definition": "//var scope = question.scope;\n  \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px',{boundingBox:[-1,limits[0]*1.2,x[x.length-1]+1,limits[1]*1.2],grid:false});\n //$(question.display.html).find('#piecewise_moment').append(div);\n  \n  var board = div.board;\n\n// eliminate duplicate points  \nArray.prototype.unique = function() {\n    return Array.from(new Set(this));\n}\n  // set up points array\n  var num_points = x.length;\n  var points = [];\n  \n//create dashed lines\nvar unique_points=x.unique();\nfor (var i=1;i<unique_points.length;i++){\n  //create vertical lines;\n    board.create('line',[[unique_points[i],0],[unique_points[i],1]],{dash:3,fixed:true,strokeWidth:0.5});\n}\n\nboard.create('point',[0,0],{fixed:true,withlabel:false});\nboard.create('point',[x[x.length-1],0],{fixed:true,withlabel:false});\nboard.create('segment',[[0,0],[0,mR[0]]],{fixed:true});\nvar p=[];\nfor (var i=0;i<x.length-1;i++){\n  p1 = board.create('point',[x[i],mR[i]],{fixed:true,withlabel:false});\n  p2 = board.create('point',[0.5*(x[i]+x[i+1]),mM[i]],{fixed:true,withlabel:false});\n  p3 = board.create('point',[x[i+1],mL[i+1]],{fixed:true,withlabel:false});\n  \n  var polyInt = JXG.Math.Numerics.lagrangePolynomial([p1,p2,p3]);\n  var gr=board.create('functiongraph',[polyInt,x[i],x[i+1]]);  \n  //board.create('segment',[p1,p2]);\n  board.create('segment',[p3,[x[i+1],mR[i+1]]]);\n}\n return div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Fill in the table of values $V(x)$&nbsp;at&nbsp;the end points of each interval. You can also hold and drag the red points on the graph.</p>\n<table border=\"1\">\n<thead>\n<tr>\n<th>Point</th>\n<th>A</th>\n<th>B left</th>\n<th>B right</th>\n<th>C left</th>\n<th>C right</th>\n<th>D</th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td>$V$, N</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n</tr>\n</tbody>\n</table>\n<div></div>\n<div>{dragpoint_board(intervals,vlimits)}</div>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "vA", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_right[0]", "maxValue": "v_right[0]", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vB_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_left[1]", "maxValue": "v_left[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vB_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_right[1]", "maxValue": "v_right[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vC_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_left[2]", "maxValue": "v_left[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vC_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_right[2]", "maxValue": "v_right[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "vD", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v_left[3]", "maxValue": "v_left[3]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Fill in the table of values for $M(x)$&nbsp;at the end and middle points of each interval. You can also hold and drag the red points on the graph.&nbsp;&nbsp;\u200b</p>\n<table border=\"1\">\n<thead>\n<tr>\n<th>Point</th>\n<th>A</th>\n<th>midle AB</th>\n<th>B left</th>\n<th>B right</th>\n<th>midle BC</th>\n<th>C left</th>\n<th>C right</th>\n<th>midle CD</th>\n<th>D</th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td>$M$, Nm</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n<td>[[8]]</td>\n</tr>\n</tbody>\n</table>\n<div></div>\n<div>{piecewise_moment(intervalsM,mlimits)}</div>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "mA", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[0]", "maxValue": "m_right[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mAB", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_middle[0]", "maxValue": "m_middle[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mB_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_left[1]", "maxValue": "m_left[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mB_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[1]", "maxValue": "m_right[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mBC", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_middle[1]", "maxValue": "m_middle[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mC_left", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_left[2]", "maxValue": "m_left[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mC_right", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_right[2]", "maxValue": "m_right[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "mCD", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m_middle[2]", "maxValue": "m_middle[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, 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